\documentclass{article}

\usepackage{amsmath,amssymb,amsthm}
\usepackage{xspace}

\title{Iteration of complex quadratic polynomials}
\author{Adrien Douady and John Hamal Hubbard}
\date{January 18, 1982}

\newcommand{\C}{\ensuremath{\mathbb{C}\xspace}}
\newcommand{\R}{\ensuremath{\mathbb{R}\xspace}}
\newcommand{\Q}{\ensuremath{\mathbb{Q}\xspace}}
\newcommand{\Log}{\operatorname{Log}}
\newcommand{\interior}[1]{\ensuremath{\overset\circ{#1}\xspace}}

\newtheorem{theorem}{Theorem}

\begin{document}
\maketitle
\section{Introduction}
...

Let $\Sigma$ denote the Riemann sphere $\C \cup \{x\}$. For all
$c\in\C$, let $f_c$ be the application of $z \mapsto z^2+c$ \emph{de
  $\Sigma$ dans elle-m\^eme et $K_c$} set of all $z\in\C$ such that
$f_c^n(z)$ goes to $\infty$ (letting $f^n$ denote the iteration
$f\circ \dotso\circ f$).  The frontier of $K_c$ is the Julia set of
$f_c$. If $0\in K_c$, then $K_c$ will be a connected set, otherwise it
will ee homeomorphic to the Cantor set (REF, REF). Let $M$ denote the
set $c\in C$ such that $0\in K_c$. Beo\^it Mandelbrot obtained the
first beautiful compter image of $M$, showing many small islands
detached from the principal component. The islands are in fact
attached by filaments which are missed by the computer:

\begin{theorem}
\label{M is connected}
The set $M$ is connected.
\end{theorem}

Let $D$ be the closed unit disk in \C. To demonstrate Theorem~\ref{M
  is connected}, construct the one analytic homeomorphism
$\Psi\colon\Sigma-D\to\Sigma-M$, tangent to the identity at $\infty$.
We shall describe this construction, then, without demonstration, a
more precise description of M. N. Sibony's proof of Theorem~\ref{M is
  connected}.

\section{Application of $\Phi$}

For all $c$, there exists a unique germ $\phi_c$, the application of
$\Sigma$ \emph{dans elle-me\^me en $\infty$}, tangen to the identity
on one point, such that $\phi_c\circ f_c\circ\phi_c^{-1}=f_0 \colon
z\mapsto z^2$.  The function $\eta_c=\Log|\phi_c|$ can be extended to
an \R-analytic function on $\C-K_c$ by
$\eta_c(z)=\lim2^{-n}\Log|f_c^n(z)|$, and in a continuous function
$\Sigma\to\R$ by $\eta_c(z)=0$ for $z\in K_c$ and
$\eta_c(\infty)=\infty$. The function $(c,z)\mapsto\eta_c(z)$ is
continuous on $\C\times\Sigma$. The other part of $\eta_c$ is a
Green's function on $K_c$.

For $c\in M$, the function $\phi_c$ extends to a analytic homeomophism
of $\Sigma-K_c$ on $\Sigma-D$. For $c\not\in M$, the function $\phi_c$
extends to a homeomorphism on $\Sigma-L_c$ or $L_c$ is a set of $z$
such that $\eta_c(z) \le \eta_c(0)$ on the complement of a disc of
radius $> 1$. The set $L_c$ is compact, limited by a curve
homeomorphic to a lemniscate, and $K_c\subset L_c$.

For $c\not\in M$ one has $\eta_c(c)=2\eta_c(0)$, \emph{done} $c\not\in
L_c$ and one can show that:
\[
\Phi(c)=\phi_c(c).
\]
One can show $\Phi(\infty)=\infty$. One can verify that $\Phi$ is
analytic on $\Sigma-M$ in $\Sigma-D$, proper of degree 1. It is there
a analytic isomophic, and one shows $\Psi=\Phi^{-1}$.

\section{Components of the Interior or $M$}
The interior \interior M of $M$ is the set of $c$ such that $f_c$
admist a point of periodic attraiction[REF], and $M$ is the
\emph{adh\'erence} of \interior M. For every component $U$ of
\interior M, one defines a function $\rho\colon U\to \interior D$ in
the following manner: it exists for every $k$ such that, for all $c\in
U$, $f_c$ admits a periodic attricting point $a$ of period $k$; as it
\emph{n'y a qu'un} attraction cycle, $(f_c^k)'(a)$ is independent of
the choice of $a$ in this cycle and one can not $p(c)$.

\begin{theorem}
For every component $U$ of \interior M, the function $p$ is a
  homeomorphism of $U$ on \interior D.
\end{theorem}

The function $\rho$ admits an extension to a homeomorphism from
$\overline U$ to $D$, analytic \emph{sauf \`eventuellement} on
$\rho^{-1}(1)$. We will call the \emph{root} of $U$ the point
$\rho^{-1}(1)$ and the \emph{center} of $U$ the point
$\rho^{-1}(0)$. The center of $U$ is the unique $c\in U$ such that
$f_c$ admits a superattractive periodic point. All $c\in\partial U$
such that $\rho(c)$ is of the form $e^{2\pi i t}$ with $t$ rational is
the root of the component $U'$ of \interior M; on says that $U'$ can
be deduced from $U$ by bifurcation.

\section{Radial Extension of $\Psi$}

It is true that $\Psi\colon \Sigma-D\to\Sigma-M$ admits a continuous
extension of $\Sigma-\interior D$ in $\Sigma$. We have only to
demonstrate the following result:

\begin{theorem}
  For all $\theta\in\Q$, the function $r\mapsto\Psi(re^{2\pi
    i\theta})$ has a limit $c_0$ when $r$ approaches $1$.
\end{theorem}
\emph{Complement} -- 
\begin{enumerate}
\item If $\theta$ is of the form $p/2^k$, for $c=c_0$ the point
  $f_c^{k+1}(0)$ is a fixed point and a repeller of $f_c$.
\item More generally, if $\theta$ is of the form $p/q$ where $q$ is
  even and $p$ is odd, in $f_{c_0}$ the point falls into a repulsive
  cycle after a finite amount of time.
\item If $\theta$ is of the forme $p/q$ with $q$ odd, the point $c_0$
  is a root of the component of \interior M.
\end{enumerate}

\section{The Tree $H_c$}
$K\subset\C$ is compact such that $\C-K$ is connected and, for all
components $U$ of \interior K, the conformal representation of $U$ on
\interior D extended to a homeomorphism of $\overline U$ on $D$. We
will say that a part $A$ of $K$ is $K$-convex if $A$ is connected and
$K\cap\overline U$ geodesically convexe (in the Poincar\'e matrix on
$U$) for all components $U$ of \interior K. The intersection of the
family of $K$-convex parts is $K$-convex, this will permit us to
define the $K$-convex envelope of on part of $K$.

As $c\in M$ such that the orbit approaches $0$ par $f_c$ is finite. This 
 can produce two \emph{fa\,cons}
\end{document}